3.809 \(\int x (c+a^2 c x^2)^{5/2} \tan ^{-1}(a x)^{3/2} \, dx\)

Optimal. Leaf size=65 \[ \frac{\left (a^2 c x^2+c\right )^{7/2} \tan ^{-1}(a x)^{3/2}}{7 a^2 c}-\frac{3 \text{Unintegrable}\left (\left (a^2 c x^2+c\right )^{5/2} \sqrt{\tan ^{-1}(a x)},x\right )}{14 a} \]

[Out]

((c + a^2*c*x^2)^(7/2)*ArcTan[a*x]^(3/2))/(7*a^2*c) - (3*Unintegrable[(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]],
x])/(14*a)

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Rubi [A]  time = 0.116419, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{3/2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2),x]

[Out]

((c + a^2*c*x^2)^(7/2)*ArcTan[a*x]^(3/2))/(7*a^2*c) - (3*Defer[Int][(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]], x
])/(14*a)

Rubi steps

\begin{align*} \int x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{3/2} \, dx &=\frac{\left (c+a^2 c x^2\right )^{7/2} \tan ^{-1}(a x)^{3/2}}{7 a^2 c}-\frac{3 \int \left (c+a^2 c x^2\right )^{5/2} \sqrt{\tan ^{-1}(a x)} \, dx}{14 a}\\ \end{align*}

Mathematica [A]  time = 6.80532, size = 0, normalized size = 0. \[ \int x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{3/2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2),x]

[Out]

Integrate[x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2), x]

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Maple [A]  time = 0.884, size = 0, normalized size = 0. \begin{align*} \int x \left ({a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}} \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^(3/2),x)

[Out]

int(x*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^(3/2),x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^(3/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a**2*c*x**2+c)**(5/2)*atan(a*x)**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}} x \arctan \left (a x\right )^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^(3/2),x, algorithm="giac")

[Out]

integrate((a^2*c*x^2 + c)^(5/2)*x*arctan(a*x)^(3/2), x)